Interactive Control of Planar Class A Bézier Curves using Logarithmic Curvature Graphs

Abstract

We present a method for interactively drawing a planar class A Bézier curve segment. First, we present a method for interactively drawing a typical class A Bézier curve segment by specifying three points like a quadratic Bézier curve segment. We show that as the degree of a typical class A Bézier curve segment is elevated, the curve converges to a logarithmic spiral segment. At the limit of infinite degree, the curve segment becomes a logarithmic spiral segment. We also present a method for drawing a general class A Bézier curve segment by perturbing the elements of the typical class A matrix so that the endpoint constraints are satisfied. To see the characteristics of the generated curves, we propose to use logarithmic curvature graphs.

Interactive Generation of 3D Class A Bézier Curve Segments

“…Note that either ofâ 1 â 0 orâ 3 â 2 is 0, the curve cannot be generated. We then generate a 2D class A Bézier curve whose endpoints are 0 a and 3 a and their tangents are 0 v and 1 v , respectively, using the method described in [13]. See Fig.…”

“…In general, the simultaneous control of the monotonicity of the curvature (and torsion for space curves) and the endpoint constraints (two endpoints and their tangents) is not easy. The papers [11,12,13,14] have addressed the problem. In case of space curves, log-aesthetic space curve In this paper, we extend our previous work of interactively controlling 2D class A Bézier curves [13] to 3D class A Bézier curves.…”

“…The papers [11,12,13,14] have addressed the problem. In case of space curves, log-aesthetic space curve In this paper, we extend our previous work of interactively controlling 2D class A Bézier curves [13] to 3D class A Bézier curves. In case of 2D class A Bézier curves, we used the geometric properties of the curves for interactive control.…”

Logarithmic Curvature and Torsion Graphs

“…Note that (13) does not guarantee the curvature and torsion are monotonically varying. A method for drawing typical 3D class A Bézier curves by specifying two endpoints and their tangents is proposed in [4].…”

“…LCGs and LTGs offer the information related to the curvature and torsion variations that the curvature and torsion plots do not offer. For planar curves, LCGs are already introduced in [12,13]. In this paper, however, we draw LCGs and LTGs from space curves and derive more detailed characteristics.…”